Quasi-Gorensteinness of extended Rees algebras. (English) Zbl 1390.13015

Kim, Youngsu

J. Commut. Algebra 9, No. 4, 507-544 (2017).

A noetherian ring with a canonical module is called quasi-Gorenstein if it is locally isomorphic to its canonical module. With this definition, a noetherian ring is Gorenstein if and only if it is Cohen-Macaulay and quasi-Gorenstein. While the quasi-Gorenstein property is generally weaker than the Gorenstein property, results of W. Heinzer et al. [J. Pure Appl. Algebra 201, No. 1–3, 264–283 (2005; Zbl 1092.13005)] showed that, in some cases, the quasi-Gorenstein property of an extended Rees algebra \(R[It,t^{-1}]\) implies that \([R[It,t^{-1}]\) is Gorenstein. They also raised the question whether this is true in much more generality. More precisely, if \((R, \mathfrak{m})\) is a local Gorenstein ring and \(I\) is an \(\mathfrak{m}\)-primary ideal, is the extended Rees algebra \(R[It,t^{-1}]\) Gorenstein if it is quasi-Gorenstein?
In this paper it is proved that the answer is yes for a special class of almost complete intersection ideals. Moreover, if \(R\) is a polynomial ring of dimension \(d\), the author shows that the conclusion holds if \(I\) is a height \(d\) monomial ideal that has a \(d\)-generated monomial reduction.
The paper contains several results regarding the core of the power ideals \(I^{n}\) and the \(\mathbf a\)-invariant of a quasi-Gorenstein extended Rees algebra \(R[It,t^{-1}]\). The author also studies the Gorenstein property of the normalization of \(R[It,t^{-1}]\) if \(I\) is a monomial ideal.

Reviewer: Catalin Ciuperca (Fargo)

MSC:

13A30	Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13H10	Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Keywords:

extended Rees algebra; associated graded ring; quasi-Gorenstein ring; Cohen-Macaulay ring

Citations:

Zbl 1092.13005